{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:29:49Z","timestamp":1772119789397,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T00:00:00Z","timestamp":1689120000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["41721003"],"award-info":[{"award-number":["41721003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["42030105"],"award-info":[{"award-number":["42030105"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["42274011"],"award-info":[{"award-number":["42274011"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["42074019"],"award-info":[{"award-number":["42074019"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["41974034"],"award-info":[{"award-number":["41974034"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["42204006"],"award-info":[{"award-number":["42204006"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (NSFC)","doi-asserted-by":"publisher","award":["2020-228"],"award-info":[{"award-number":["2020-228"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Space Station Project","award":["41721003"],"award-info":[{"award-number":["41721003"]}]},{"name":"Space Station Project","award":["42030105"],"award-info":[{"award-number":["42030105"]}]},{"name":"Space Station Project","award":["42274011"],"award-info":[{"award-number":["42274011"]}]},{"name":"Space Station Project","award":["42074019"],"award-info":[{"award-number":["42074019"]}]},{"name":"Space Station Project","award":["41974034"],"award-info":[{"award-number":["41974034"]}]},{"name":"Space Station Project","award":["42204006"],"award-info":[{"award-number":["42204006"]}]},{"name":"Space Station Project","award":["2020-228"],"award-info":[{"award-number":["2020-228"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>We introduce an approach for the direct measurement of the gravitational potential (GP) along the trajectory of a satellite, with a specific focus on Low-Earth Orbit (LEO) satellites. A LEO satellite communicates with several Geosynchronous Equatorial Orbit (GEO) satellites via frequency signal links. The GP difference can be measured in real-time using the gravitational frequency shift approach by equipping both LEO and GEO satellites with precise atomic clocks. Since the GP at the high orbits of the GEO satellites can be precisely determined by the present gravitational field model EGM2008, the GP along the LEO satellite\u2019s trajectory can be determined. In this study, simulation experiments were conducted, featuring a GRACE-type satellite as the LEO satellite in communication with three equidistant GEO satellites. The results indicated that the accuracy of the GP measurements along the LEO satellite\u2019s trajectory primarily depends on the precision of the onboard atomic clocks. Supposing optical atomic clocks attain an instability level of 1\u00d710\u221217\u03c4\u22121\/2 (\u03c4 in seconds), we determined the GP distribution covered by the LEO satellite\u2019s trajectories with 30-day observations. Then, we determined a gravitational field at the centimeter level based on the GP distribution. The GP data derived from the trajectory of a LEO satellite can be utilized to establish temporal gravitational fields, which have broad applications in different disciplines.<\/jats:p>","DOI":"10.3390\/rs15143514","type":"journal-article","created":{"date-parts":[[2023,7,13]],"date-time":"2023-07-13T01:52:25Z","timestamp":1689213145000},"page":"3514","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Method for Measuring Gravitational Potential of Satellite\u2019s Orbit Using Frequency Signal Transfer Technique between Satellites"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7261-1068","authenticated-orcid":false,"given":"Ziyu","family":"Shen","sequence":"first","affiliation":[{"name":"School of Resource, Environmental Science and Engineering, Hubei University of Science and Technology, Xianning 437100, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9267-5982","authenticated-orcid":false,"given":"Wenbin","family":"Shen","sequence":"additional","affiliation":[{"name":"Time and Frequency Geodesy Center, Department of Geophysics, School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1614-1900","authenticated-orcid":false,"given":"Xinyu","family":"Xu","sequence":"additional","affiliation":[{"name":"Time and Frequency Geodesy Center, Department of Geophysics, School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China"}]},{"given":"Shuangxi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Time and Frequency Geodesy Center, Department of Geophysics, School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6194-9661","authenticated-orcid":false,"given":"Tengxu","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Resource, Environmental Science and Engineering, Hubei University of Science and Technology, Xianning 437100, China"}]},{"given":"Lin","family":"He","sequence":"additional","affiliation":[{"name":"School of Resource, Environmental Science and Engineering, Hubei University of Science and Technology, Xianning 437100, China"}]},{"given":"Zhan","family":"Cai","sequence":"additional","affiliation":[{"name":"School of Resource, Environmental Science and Engineering, Hubei University of Science and Technology, Xianning 437100, China"}]},{"given":"Si","family":"Xiong","sequence":"additional","affiliation":[{"name":"School of Resource, Environmental Science and Engineering, Hubei University of Science and Technology, Xianning 437100, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1212-023X","authenticated-orcid":false,"given":"Lingxuan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Resource, Environmental Science and Engineering, Hubei University of Science and Technology, Xianning 437100, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/0031-9201(81)90046-7","article-title":"Preliminary reference Earth model","volume":"25","author":"Dziewonski","year":"1981","journal-title":"Phys. Earth Planet. Inter."},{"key":"ref_2","unstructured":"Hofmann-Wellenhof, B., and Moritz, H. (2005). Physical Geodesy, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kaula, W.M. (1966). Theory of Satellite Geodesy: Applications of Satellites to Geodesy, Dover Publications.","DOI":"10.1063\/1.3033941"},{"key":"ref_4","first-page":"285","article-title":"The CHAMP geopotential mission","volume":"40","author":"Reigber","year":"1999","journal-title":"Boll. Geof. Teor. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"L09607","DOI":"10.1029\/2004GL019920","article-title":"The gravity recovery and climate experiment: Mission overview and early results","volume":"31","author":"Tapley","year":"2004","journal-title":"Geophys. Res. Lett."},{"key":"ref_6","unstructured":"Drinkwater, M.R., Haagmans, R., Muzi, D., Popescu, A., Floberghagen, R., Kern, M., and Fehringer, M. (2006, January 6\u20138). The GOCE gravity mission: ESA\u2019s first core Earth explorer. Proceedings of the 3rd International GOCE User Workshop, Frascati, Italy."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1007\/s00190-003-0362-1","article-title":"A technique for modeling the Earth\u2019s gravity field on the basis of satellite accelerations","volume":"78","author":"Ditmar","year":"2004","journal-title":"J. Geod."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s001900100159","article-title":"Gravity recovery using COSMIC GPS data: Application of orbital perturbation theory","volume":"75","author":"Hwang","year":"2001","journal-title":"J. Geod."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s00190-003-0322-9","article-title":"Harmonic analysis of the Earth\u2019s gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP","volume":"77","author":"Reubelt","year":"2003","journal-title":"J. Geod."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1023\/A:1008313405488","article-title":"The determination of gravitational potential differences from satellite-to-satellite tracking","volume":"75","author":"Jekeli","year":"1999","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"36-1","DOI":"10.1029\/2002GL015180","article-title":"Efficient gravity field recovery using in situ disturbing potential observables from CHAMP","volume":"29","author":"Han","year":"2002","journal-title":"Geophys. Res. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s00190-003-0315-8","article-title":"Energy integral method for gravity field determination from satellite orbit coordinates","volume":"77","author":"Visser","year":"2003","journal-title":"J. Geod."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1038\/s41566-019-0493-4","article-title":"Demonstration of 4.8 \u00d7 10\u221217 stability at 1 s for two independent optical clocks","volume":"13","author":"Oelker","year":"2019","journal-title":"Nat. Photonics"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1126\/science.abb2473","article-title":"Coherent optical clock down-conversion for microwave frequencies with 10\u201318 instability","volume":"368","author":"Nakamura","year":"2020","journal-title":"Science"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1038\/s41586-021-04344-y","article-title":"Differential clock comparisons with a multiplexed optical lattice clock","volume":"602","author":"Zheng","year":"2022","journal-title":"Nature"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"053204","DOI":"10.1063\/1.5090583","article-title":"Towards a transportable aluminium ion quantum logic optical clock","volume":"90","author":"Hannig","year":"2019","journal-title":"Rev. Sci. Instrum."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10291-020-01059-x","article-title":"Past, present and future of atomic clocks for GNSS","volume":"25","author":"Jaduszliwer","year":"2021","journal-title":"GPS Solut."},{"key":"ref_18","first-page":"844","article-title":"Die Feldgleichungen der Gravitation","volume":"25","author":"Einstein","year":"1915","journal-title":"Sitzungsberichte K\u00f6niglich Preuss. Akad. Wiss."},{"key":"ref_19","first-page":"388","article-title":"Determination of the geopotential and orthometric height based on frequency shift equation","volume":"3","author":"Shen","year":"2011","journal-title":"Nat. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1162","DOI":"10.1093\/gji\/ggw198","article-title":"Formulation of geopotential difference determination using optical-atomic clocks onboard satellites and on ground based on Doppler cancellation system","volume":"206","author":"Shen","year":"2016","journal-title":"Geophys. J. Int."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1007\/s10712-017-9414-6","article-title":"Determination of gravitational potential at ground using optical-atomic clocks on board satellites and on ground stations and relevant simulation experiments","volume":"38","author":"Shen","year":"2017","journal-title":"Surv. Geophys."},{"key":"ref_22","unstructured":"Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley."},{"key":"ref_23","first-page":"207","article-title":"On a relativistic geodesy","volume":"59","author":"Bjerhammar","year":"1985","journal-title":"Bull. Am. Assoc. Hist. Nurs."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"012051","DOI":"10.1088\/1742-6596\/723\/1\/012051","article-title":"Relativistic geodesy","volume":"723","author":"Flury","year":"2016","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Puetzfeld, D., and L\u00e4mmerzahl, C. (2019). Relativistic Geodesy: Foundations and Applications, Springer.","DOI":"10.1007\/978-3-030-11500-5"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1134\/S0202289316030099","article-title":"Chronometric measurement of orthometric height differences by means of atomic clocks","volume":"22","author":"Kopeikin","year":"2016","journal-title":"Gravit. Cosmol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1038\/s41567-017-0042-3","article-title":"Geodesy and metrology with a transportable optical clock","volume":"14","author":"Grotti","year":"2018","journal-title":"Nat. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"662","DOI":"10.1038\/nphoton.2016.159","article-title":"Geopotential measurements with synchronously linked optical lattice clocks","volume":"10","author":"Takano","year":"2016","journal-title":"Nat. Photonics"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1594","DOI":"10.1093\/gji\/ggy508","article-title":"Clock networks for height system unification: A simulation study","volume":"216","author":"Wu","year":"2019","journal-title":"Geophys. J. Int."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1007\/s12583-018-0834-0","article-title":"Formulation of determining the gravity potential difference Using ultra-high precise clocks via optical fiber frequency transfer technique","volume":"30","author":"Shen","year":"2019","journal-title":"J. Earth Sci."},{"key":"ref_31","first-page":"021016","article-title":"Synchronization of distant optical clocks at the femtosecond Level","volume":"6","author":"Sinclair","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/BF00653616","article-title":"An orbiting clock experiment to determine the gravitational red shift","volume":"6","author":"Kleppner","year":"1970","journal-title":"Astrophys. Space Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/BF00759854","article-title":"A test of the equivalence principle using a space-borne clock","volume":"10","author":"Vessot","year":"1979","journal-title":"Gen. Relat. Grav."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2081","DOI":"10.1103\/PhysRevLett.45.2081","article-title":"Test of relativistic gravitation with a space-borne hydrogen maser","volume":"45","author":"Vessot","year":"1980","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s00190-020-01401-8","article-title":"Using quantum optical sensors for determining the Earth\u2019s gravity field from space","volume":"94","author":"Wu","year":"2020","journal-title":"J. Geod."},{"key":"ref_36","unstructured":"Shen, Z., Shen, W., and Zhang, S. (2018, January 8\u201313). Determination of the gravitational potential at GOCE-type satellite orbit using frequency signal transmission approach. Proceedings of the 20st EGU General Assembly, EGU2018, Vienna, Austria."},{"key":"ref_37","unstructured":"Pierno, L., and Varasi, M. (2013). Switchable Delays Optical Fibre Transponder with Optical Generation of Doppler Shift. (8,466,831), U.S. Patent."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1007\/BF02525647","article-title":"Physical geodesy","volume":"86","author":"Heiskanen","year":"1967","journal-title":"Bull. Geod."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2760","DOI":"10.1038\/s41467-018-05219-z","article-title":"In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms","volume":"9","author":"Liu","year":"2018","journal-title":"Nat. Commun."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"B04406","DOI":"10.1029\/2011JB008916","article-title":"The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)","volume":"117","author":"Pavlis","year":"2012","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"423","DOI":"10.19062\/2247-3173.2016.18.1.58","article-title":"Satellite tracking using norad two-line element set format","volume":"18","author":"Croitoru","year":"2016","journal-title":"AFASES"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1002\/2016SW001593","article-title":"International reference ionosphere 2016: From ionospheric climate to real-time weather predictions","volume":"15","author":"Bilitza","year":"2017","journal-title":"Space Weather"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/BF01036419","article-title":"Doppler frequency shift during ionospheric propagation of decameter radio waves","volume":"18","author":"Namazov","year":"1975","journal-title":"Radiophys. Quantum Electron."},{"key":"ref_44","unstructured":"Voigt, C., F\u00f6rste, C., Wziontek, H., Crossley, D., Meurers, B., P\u00e1link\u00e1\u0161, V., Hinderer, J., Boy, J.P., Barriot, J.P., and Sun, H. (2017, January 23\u201328). The Data Base of the International Geodynamics and Earth Tide Service (IGETS). Proceedings of the 19th EGU General Assembly, EGU2017, Vienna, Austria."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1016\/j.cageo.2004.11.015","article-title":"Tsoft: Graphical and interactive software for the analysis of time series and Earth tides","volume":"31","author":"Vauterin","year":"2005","journal-title":"Comput. Geosci."},{"key":"ref_46","first-page":"9425","article-title":"The nanogal software: Earth tide data processing package ETERNA 3.30","volume":"124","author":"Wenzel","year":"1996","journal-title":"Bull. Inf. Mar\u00e9es Terrestres"},{"key":"ref_47","unstructured":"Joernc (2023, March 10). Tidal-Potential. Available online: https:\/\/github.com\/joernc\/tidal-potential."},{"key":"ref_48","unstructured":"Major, F.G. (2013). The Quantum Beat: The Physical Principles of Atomic Clocks, Springer Science & Business Media."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"S257","DOI":"10.1088\/0026-1394\/40\/3\/305","article-title":"A mathematical model for the atomic clock error","volume":"40","author":"Galleani","year":"2003","journal-title":"Metrologia"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"103","DOI":"10.2478\/arsa-2013-0009","article-title":"A comparison between numerical differentiation and kalman filtering for a leo satellite velocity determination","volume":"48","author":"Sharifi","year":"2013","journal-title":"Artif. Satell."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Li, X., Zhu, Y., Zheng, K., Yuan, Y., Liu, G., and Xiong, Y. (2020). Precise Orbit and Clock Products of Galileo, BDS and QZSS from MGEX Since 2018: Comparison and PPP Validation. Remote Sens., 12.","DOI":"10.3390\/rs12091415"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Shen, P., Cheng, F., Lu, X., Xiao, X., and Ge, Y. (2021). An Investigation of Precise Orbit and Clock Products for BDS-3 from Different Analysis Centers. Sensors, 21.","DOI":"10.3390\/s21051596"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1029\/JZ067i002p00601","article-title":"A study of F 2 -layer effects as observed with a Doppler technique","volume":"67","author":"Davies","year":"1962","journal-title":"J. Geophys. Res."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s00190-006-0106-0","article-title":"Higher order ionospheric effects in precise GNSS positioning","volume":"81","author":"Hoque","year":"2007","journal-title":"J. Geod."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"2159","DOI":"10.5194\/angeo-24-2159-2006","article-title":"On the validity of the ionospheric pierce point (IPP) altitude of 350 km in the Indian equatorial and low-latitude sector","volume":"24","author":"Niranjan","year":"2006","journal-title":"Ann. Geophys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1214\/aos\/1176343414","article-title":"Extension of the Gauss-Markov theorem to include the estimation of random effects","volume":"4","author":"Harville","year":"1976","journal-title":"Ann. Statist."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/14\/3514\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:11:35Z","timestamp":1760127095000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/14\/3514"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,12]]},"references-count":56,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["rs15143514"],"URL":"https:\/\/doi.org\/10.3390\/rs15143514","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,12]]}}}